Suppose we have a sequence $(a_n)$ of real numbers and we are interested in the existence and value of the series
\begin{equation}
\lim_{N \to \infty} \qquad \sum^N_{n = 1} (a_n - a_{n-1}) \,.
\end{equation}
Looking at the partial sum, we have $S_N = a_N - a_0$.

1 Answer
1

I thought I answered this at the other question. $T_N$ is not a rearrangement of $S_N$. Pick a value of $N$, and write out $S_N$, and write out $T_N$, and you will see that there are terms in the one that are not in the other.

Ah! Now I understand, this is where my confusion comes from, because the example actually talks about different sequences of partial sums. I am sorry I didn't get your answer in the first place!
–
harlekinJan 22 '12 at 12:13